Waveform gap filling

ABSTRACT

A method for filling a gap of missing vibration data in a set of vibration data. At least one reference waveform on a first side of the gap is selected and at least one adjacent waveform on an opposing second side of the gap is selected. It is determined whether the gap is in a section of the vibration data where a frequency of the vibration data is one of increasing, decreasing, and steady state. Where the gap is in a section of the vibration data where the frequency of the vibration data is changing substantially linearly, an analytical method is applied to at least one of the at least one reference waveform and the at least one adjacent waveform to approximate the vibration data that is missing in the gap. Where the gap is in a section of the vibration data where the frequency of the vibration data is changing substantially exponentially, a numerical method is applied to at least one of the at least one reference waveform and the at least one adjacent waveform to approximate the vibration data that is missing in the gap. Where the gap is in a section of the vibration data where the frequency of the vibration data is substantially steady state, at least one of the at least one reference waveform and the at least one adjacent waveform is copied to approximate the vibration data that is missing in the gap. The approximated vibration data is presented to a user.

FIELD

This invention relates to the field of data analysis. More particularly,this invention relates to creating estimated waveforms that can be usedto fill in gaps of missing data in vibration data sets.

INTRODUCTION

Machine vibration information, such as might be gathered on rotatingmachines like a turbine, can be very useful when monitoring the healthof the machine. For example, many machine problems can be diagnosed froman analysis of the vibration information. This information is typicallyrepresented as a sinusoidal waveform, with a frequency and amplitudethat increase as the rotational speed of the machine increases.

Because such machines generally rotate at a relatively high speed, alarge amount of data must typically be gathered to adequately representthe vibrational characteristics of the machine, and to capture theinformation that would be required to diagnose any issues with themachine. Because of the large amount of data, it is problematic to bothtransmit the data and to store the data. Transmission problems arisebecause of the bandwidth that would be required by the large amount ofdata, and storage problems arise because of the amount of storage spacethat would be required by the large amount of data.

In order to reduce such problems, some vibration instruments only passalong a subset of the vibration data. This can be accomplished in one ormore of a variety of different ways, for example, by either not sensinga portion of the data or by not passing along a portion of the data thatis sensed. This tends to reduce the bandwidth and storage requirements.

However, it can be disconcerting to see such gaps in the data when itcomes time to analyze or otherwise review the data.

What is needed therefore, is a system for reducing issues such as thosedescribed above, at least in part.

SUMMARY

The above and other needs are met by a method for filling a gap ofmissing vibration data in a set of vibration data. At least onereference waveform on a first side of the gap is selected and at leastone adjacent waveform on an opposing second side of the gap is selected.It is determined whether the gap is in a section of the vibration datawhere a frequency of the vibration data is one of increasing,decreasing, and steady state. Where the gap is in a section of thevibration data where the frequency of the vibration data is changingsubstantially linearly, an analytical method is applied to at least oneof the at least one reference waveform and the at least one adjacentwaveform to approximate the vibration data that is missing in the gap.Where the gap is in a section of the vibration data where the frequencyof the vibration data is changing substantially exponentially, anumerical method is applied to at least one of the at least onereference waveform and the at least one adjacent waveform to approximatethe vibration data that is missing in the gap. Where the gap is in asection of the vibration data where the frequency of the vibration datais substantially steady state, at least one of the at least onereference waveform and the at least one adjacent waveform is copied toapproximate the vibration data that is missing in the gap. Theapproximated vibration data is presented to a user.

In some embodiments according to this aspect of the invention, when thegap is in a section of the vibration data where the frequency of thevibration data is substantially changing, the at least one referencewaveform has a frequency that is slower than the at least one adjacentwaveform. In some embodiments, the step of presenting the approximatedvibration data includes displaying a plot of the set of vibration datato the user with the gap filled.

In some embodiments, the set of vibration data is sensed from a rotatingmachine. In some embodiments, the set of vibration data is sensed from aturbine. In some embodiments, the frequency of the vibration data ischanging substantially linearly as a rotating speed of the turbine isincreased to an operating speed. In some embodiments, the frequency ofthe vibration data is changing substantially exponentially as a rotatingspeed of the turbine is decreased from an operating speed. In someembodiments, the frequency of the vibration data is substantially steadystate as the turbine is running at an operating speed.

In some embodiments, the step of approximating the data when usingeither the numerical method or the analytical method includes the stepsof approximating a frequency rate of change of the missing vibrationdata in the gap, approximating tach signal locations of the missingvibration data in the gap, approximating an amplitude rate of change ofthe missing vibration data in the gap, and approximating a phase rate ofchange of the missing vibration data in the gap. Then, the gap is filledwith ideal waveforms between the tach signal locations, and the idealwaveforms are adjusted using the approximated frequency, amplitude, andphase rates of change.

DRAWINGS

Further advantages of the invention are apparent by reference to thedetailed description when considered in conjunction with the figures,which are not to scale so as to more clearly show the details, whereinlike reference numbers indicate like elements throughout the severalviews, and wherein:

FIG. 1 is a graph of turning speed versus time for a rotating machine,showing a first section of linear speed increase when the machine isturned on, a second section of steady state speed during operation ofthe machine, and a third section of exponential speed decay when themachine is turned off.

FIG. 2 is a flowchart for the overall method for estimating missingwaveforms, according to an embodiment of the present invention.

FIG. 3 is a functional block diagram of an apparatus for estimatingwaveforms, according to an embodiment of the present invention.

FIG. 4 is a diagrammatic representation of the basic method forestimating missing waveforms, according to an embodiment of the presentinvention.

FIG. 5 is a flowchart of a first part of a detailed method forestimating waveforms for linear speed increases, according to anembodiment of the present invention.

FIG. 6 is a flowchart of a second part of a detailed method forestimating waveforms for linear speed increases, according to anembodiment of the present invention.

FIG. 7 is a flowchart of a third part of a detailed method forestimating waveforms for linear speed increases, according to anembodiment of the present invention.

FIG. 8 is a flowchart of a fourth part of a detailed method forestimating waveforms for linear speed increases, according to anembodiment of the present invention.

FIG. 9 is a flowchart of a fifth part of a detailed method forestimating waveforms for linear speed increases, according to anembodiment of the present invention.

FIG. 10A is a flowchart of a sixth part of a detailed method forestimating waveforms for linear speed increases, according to anembodiment of the present invention.

FIG. 10B is a flowchart of a seventh part of a detailed method forestimating waveforms for linear speed increases, according to anembodiment of the present invention.

FIG. 11 is a flowchart of a first part of a detailed method forestimating waveforms for exponential speed decay, according to anembodiment of the present invention.

FIG. 12 is a flowchart of a second part of a detailed method forestimating waveforms for exponential speed decay, according to anembodiment of the present invention.

FIG. 13 is a flowchart of a third part of a detailed method forestimating waveforms for exponential speed decay, according to anembodiment of the present invention.

FIG. 14 is a flowchart of a fourth part of a detailed method forestimating waveforms for exponential speed decay, according to anembodiment of the present invention.

FIG. 15 is a flowchart of a fifth part of a detailed method forestimating waveforms for exponential speed decay, according to anembodiment of the present invention.

FIG. 16A is a flowchart of a sixth part of a detailed method forestimating waveforms for exponential speed decay, according to anembodiment of the present invention.

FIG. 16B is a flowchart of a seventh part of a detailed method forestimating waveforms for exponential speed decay, according to anembodiment of the present invention.

FIG. 17A depicts a first representation of a reference waveform, gap,and adjacent waveform.

FIG. 17B depicts a second representation of a reference waveform, gap,and adjacent waveform.

DESCRIPTION

With reference now to FIG. 1, there is depicted a representational graph100 of turning speed versus time for a rotating machine, with a waveform102. As depicted, there is a first section 104 of linear speed increasewhen the machine is turned on, a second section 106 of steady statespeed during operation of the machine, and a third section 108 ofexponential speed decay when the machine is turned off.

Although not indicated on the graph, the relative lengths of time forthese three sections 104, 106, and 108 might not be as depicted. Forexample, it might take several hours for the machine to be ramped up toits operating speed, as represented by section 104, and then it mightremain at the operating speed for weeks or months, as represented bysection 106. Finally, it might take only a matter of minutes or a coupleof hours for the machine to stop rotating once it is turned off, asrepresented by section 108.

As depicted in FIGS. 17A and 17B, the waveform data provided has gaps1702 in it, so as to reduce transmission, storage, and other issuesassociated with the amount of data that is produced. Instead ofrecording, transmitting, and storing actual vibration data for thesegaps 1702, some embodiments of the present invention fill any such gaps1702 with estimates of the waveform 102 data. These estimates areproduced by analyzing the waveform 102 data this is provided on eitherside of the gap 1702 that is to be filled.

In the second section 106 of the graph 100, re-creation of waveforms 102in the gaps 1702 is a relatively straight-forward procedure. Because themachine is operating at steady state, each waveform 102 in section 106looks substantially like any other waveform 102 in section 106, and sothe waveform gaps 1702 in the second section 106 can be filled in withwaveforms 102 that are either identical to those waveforms 102 thatprecede or follow the gap 1702, or are simple linear interpolations ofsuch.

It is appreciated that any anomaly that might have occurred during sucha gap 1702 cannot be recreated, regardless of the section 104, 106, or108 in which it may have occurred, as the data to do so is probably notpresent in the waveform 102 data preceding or following the gap 1702 inwhich the anomaly occurred. Thus, the waveforms 102 produced accordingto the present embodiments would tend to not be useful for investigationof any such. Instead, they tend to provide a continuous representationof data that might be useful in certain circumstances.

Estimations of the waveforms 102 to fill the gaps 1702 in the firstsection 104 and the third section 108, however, are not asstraightforward as the estimation of the waveforms 102 to fill the gaps1702 in the second section 106, because of the change in turning speedin those sections 104 and 108.

While it might seem that a simple interpolation between preceding andfollowing waveforms 102 could be performed to produce waveforms 102 forthe gaps 1702 in the first section 104 that represents a linear increasein speed, such a method is inadequate because the speed does notincrease just between successive rotations in a stepped manner. Instead,the speed increases continually throughout a given rotation, andtherefore the waveform 102 produced during a given rotation changesthroughout its period.

This problem is compounded further for the waveforms 102 that areproduced during the third section 108, because the speed of the machineis decreasing substantially exponentially instead of substantiallylinearly.

Thus, according to the embodiments described below, a quadraticanalytical solution is described in regard to estimating missingwaveforms 102 in the gaps 1702 in the first section 104, and aniterative numerical solution is described in regard to estimatingmissing waveforms 102 in the gaps 1702 in the third section 108.

In each case, both the analytical solution for the linear first section104 and the numerical solution for the exponential third section 108, acommon basic approach 200 is used, as depicted in FIG. 2. As can beseen, the waveform 102 data, with gaps 1702, is input into acomputational apparatus (such as depicted in FIG. 3) as given in block202. A gap 1702 in the data is identified, as given in block 204, andthe two waveforms 1704 and 1706 on either side of the gap 1702 areselected for further processing, as given in block 206.

It is then determined what section of the graph 100 the two selectedwaveforms belong to—be it the first section 104, the second section 106,or the third section 108, as given in block 208 of the flowchart 200.

In some embodiments, the waveform 102 next to the gap 1702 that has theslower turning speed is designated the reference waveform 1704, asdepicted in FIGS. 17A and 17B, and the waveform 102 adjacent the gap1702 that has the higher turning speed is designated the adjacentwaveform 1706. In the first section 104, the reference waveform 1704 isthe waveform 102 preceding the gap 1702 in time and the adjacentwaveform 1706 is the waveform 102 following the gap 1702 in time, asdepicted in FIG. 17A. In the third section 108, the reference waveform1704 is the waveform 102 following the gap 1702 in time and the adjacentwaveform 1706 is the waveform 102 preceding the gap 1702 in time, asdepicted in FIG. 17B.

The reason for doing this is that there are more data points per unit oftime in a waveform 102 that has a slower turning speed, and thus theaccuracy of the computations described herein can be improved by makingthe designation as described in the paragraph above. However, in otherembodiments, some other method besides just turning speed is used todetermine which waveform 102 is designated as the reference waveform1704 and the adjacent waveform 1706.

It is appreciated that the waveforms 1704 and 1706 and the gap 1702 arenot meant to imply any kind of exact relative time scale. In general,the reference waveform 1704 will have a longer period T_(r) than theperiod T_(w) of the adjacent waveform 1706. The gap 1702 with a durationof T_(g) might have one or many missing waveforms 102 that are to beestimated.

If both the reference waveform 1704 and the adjacent waveform 1706 arefrom the second section 106, then the method falls to block 210, where acopy or simple linear interpolation is performed to produce the missingwaveforms 102 and fill in the gap 1702. If both the reference 1704 andadjacent 1706 waveforms are from the first section 104, then the methodfalls to block 212, where an analytical quadratic estimation isperformed to produce the missing waveforms 102 and fill in the gap 1702.Finally, If the reference waveform 1704 and adjacent waveform 1706 arefrom the third section 108, then the method falls to block 214, where aniterative numerical estimation is performed to produce the missingwaveforms 102 and fill in the gap 1702.

The goals of the individual steps for both the analytical solution 212and the iterative numerical solution 214 are generally the same, and soare depicted in FIG. 2 with common names. However, the manner in whichthese steps are performed in the analytical solution 212 are differentfrom the manner in which the steps are performed in the iterativenumerical solution 214, as described in more detail hereafter.Therefore, the reference numbers used for each step are the same foreach method 212 and 214, but are distinguished one from another with thelabel “a” for the steps of the analytical solution 212, and “b” for thesteps of the numerical solution 214.

The general steps for both the analytical solution 212 and the iterativenumerical solution 214 proceed as follows. The frequency and speed rateof change of the missing waveforms 102 in the gap 1702 are estimated, asgiven in methods 216 a and 216 b, and the rate of speed change in thegap 1702 is adjusted, as given in methods 218 a and 218 b. Tachometerpulses are created and placed in proper positions in the gap 1702, asgiven in methods 220 a and 220 b. The change in rate of the amplitude isestimated, as given in methods 222 a and 222 b, and the rate of changeof the phase is estimated, as given in methods 224 a and 224 b. The gap1702 is filled with an estimated waveform 102 that has been adjusted forrotational speed, amplitude, and phase changes, as given in methods 226a and 226 b.

It is again mentioned that, even though these steps have similar namesin each of the analytical solution 212 and the iterative numericalsolution 214, they are performed in different ways, as described in moredetail below.

With reference now to FIG. 3, there is depicted an instrument 300 forfilling waveforms 102 into a gap 1702. In one embodiment, the instrument300 is a computer that receives the waveform 102 data through theinput/output 310, or otherwise as described herein. In otherembodiments, the instrument 300 receives all of the waveform 102 datadirectly from a sensor 308. In some embodiments the steps of the methodsas described herein are embodied in a computer language on anon-transitory storage medium 314 that is readable by the instrument 300of FIG. 3, and which enable the instrument 300 to fill the gap 1702 asdescribed herein. In some embodiments the instrument 300 includes a readonly memory 304, such as might contain basic operating instructions, anda random access memory 306, such as is used to hold data used in or forthe various computations as described herein. An interface 312 is usedfor an operator to communicate with the instrument 300.

The various embodiments of the present invention thus improve the basicoperation of an apparatus 300 as depicted. If bandwidths of the I/O 310,for example, and the storage capacity of the medium 314 could handle theextreme amounts of data that are produced by a sensor 308, then theremight be no need to only transmit and store a portion of the waveformdata, and thus there might also be no gaps 1702 in the waveform 102 datathat need to be filled.

Conceptual Overview

The first three steps 216 a-220 a and 216 b-220 b of each method 212 and214 have a goal of placing estimated tachometer signal pulses within thegap 1702, where two consecutive tach pulses t1 and t2 of the referencewaveform 1704, as depicted in FIGS. 17A and 17B, define between them onerotation of the machine from which the data was gathered. If the tachpulses in the original data signal have been preserved for the gaps1702, then no estimation of the tach pulse time locations is required.The tach pulse data might be preserved, for example, because it is arelatively small component of the overall data stream, and thus it doesnot add significantly to the overhead to retain it in the data stream.However, if the tach signal timing data has not been retained in thedata stream, then it is estimated as described herein. When therotational speed changes either linearly or exponentially as in sections104 and 108, respectively, then the time between the tach pluses changesfrom tach pulse to tach pulse.

Once the tach pulses have been placed in the gap 1702, the rate ofchange of both amplitude and phase are calculated, and then estimatedwaveforms 102 are produced in the gap 1702. FIG. 4 diagrammaticallydepicts the basic method for estimating waveforms.

The waveform 404 within the reference block 1704 is defined by twoconsecutive tach signals t₁ 424 and t₂ 426. The waveform 404 includesthe data gathered for one revolution of the machine. It is appreciatedthat the shape of the waveform 404 is representational, and in actualuse might have a different and more complex shape.

Section 1702 is the gap 1702 to be filled. It is defined by the tachsignal t₂ 426 at the end of the reference waveform block 1704 precedingthe gap 1702, and the estimated (or actual) tach signal 428 at the startof the adjacent waveform 1706.

A pure sine wave 412 is placed within both sections 1704 and 1702. Thesine wave 412 is given the same amplitude in both sections 1704 and1702, but the frequency of the sine wave 412 is different in eachsection 1704 and 1702. In section 1704 the frequency of the sine wave412 is set to exactly match one revolution between the two tach signalst₁ 424 and t₂ 426. In section 1702 the frequency of the sine wave 412 isset to exactly match one revolution between the two tach signals 426 and428.

FIG. 4 depicts a case that is taken from the first section 104 of thechart 100 in FIG. 1, and thus speed is increasing linearly. Thus, tachsignals t₁ 424 and t₂ 426 are closer to each other than are tach signals424 and 426, and therefore the frequency of the sine wave 412 in section1702 is higher than the frequency of the sine wave 412 in section 1704.

It is appreciated that if FIG. 4 depicted a case that was taken from thethird section 108 of the chart 100 in FIG. 1, then speed would bedecreasing exponentially, and tach signals t₁ 424 and t₂ 426 would befarther apart from each other than tach signals 424 and 426.

At this point, we have two sections 1704 and 1702 that are each filledwith parts of a sine wave 412, where the frequency of the sine wave 412in section 1704 is different than the frequency of the sine wave 412 insection 1702.

A position is identified at time 414 in section 1702. In someembodiments, time 414 represents the first of a predetermined number oftime divisions that will be used in each waveform 408 section (betweensuccessive tach signals) to be filled in the gap 1702. In such anembodiment, a given number of time divisions will be used for eachwaveform 408 in the gap 1702, no matter how short the time durationbetween two successive tach signals. In other embodiments, time 414represents a specific length of time. In this embodiment, shorterwaveforms in the gap 1702 (smaller distance between tach signals) areestimated using fewer time data points. In other embodiments, acombination of these two embodiments or some other method is used todetermine the times 414.

The amplitude 416 of the sine wave 412 is determined at time 414. Thisamplitude is mapped over to the corresponding time position 418 thatintersects the sine wave 412 within section 1704. Then the amplitude 420of the actual waveform 404 is determined at that time position 418.Finally, the amplitude 420 is mapped over to position 422, whichposition 422 has the amplitude 420 of the actual waveform from section1704, but at the time 414 in section 1702. By repeating this four-stepprocess for each of the desired number of time positions 414, a completeapproximated waveform 408 is constructed within the gap 1702. Using thissame process, additional waveforms 408 are constructed betweensuccessive tach signals within the gap 1702, until the gap 1702 iscompletely filled.

These waveforms 408 are corrected for amplitude, as given in step 222 ofFIG. 2. As a machine changes rotational speed, the amplitude of thevibration waveform 102 typically changes in a similar way. Therefore,some embodiments of the present invention assume that if the change inrotational speed is linear, as in section 104 of graph 100, then thechange in waveform amplitude is also linear, but at a different rate ofchange than the rotational speed rate of change. Similarly, if thechange in rotational speed is exponential (as in section 108 of graph100), then the change in waveform amplitude is assumed to beexponential.

In addition, the phase associated with the first order turning speedchanges from one waveform 102 to the next as the rotational speed of themachine changes. Therefore, the waveforms 102 that fill the gap 1702 areadjusted for this change in phase. The rate of change in phase isassumed to be either linear or exponential, according to the section 104or 108 in which the gap 1702 is identified.

The steps as generally described in this section can be accomplishedusing analytical methods for the case of the linear change in speed, asdepicted in section 104 of FIG. 1, and are performed using numericalmethods for the case of the exponential change in speed, as depicted insection 108 of FIG. 1.

Analytical Solution for Linear Portion

A. Linear Frequency and Speed

The linear rate of change in rotational speed is initially calculatedfrom the last two waveform cycles in the reference block 1704, accordingto the method 216 a as depicted in FIG. 5. This is then adjusted toensure that the tachometer pulse at the end of the gap 1702 matches upwith the first tachometer pulse in the adjacent block 1706, according tothe method 218 a as depicted in FIG. 6.

The calculations are based on the times of the last three tach pulsest₀, t₁, and t₂, in reference block 1704, as depicted in FIGS. 17A and17B. By “last” it is understood to mean the three tach pulses closest tothe gap 1702. So, those three tach pulses are identified as given inblock 504 of FIG. 5, and the times at which they exist are derived asgiven in block 506. The times are relative at this point, with the timeof the tach pulse that is furthest from the gap 1702 being given adesignation of t₀=0, and the times of the other two tach pulsespreceding sequentially in a positively increasing direction from t₁ tot₂.

A linear increase in rotational speed or frequency uses the followingequation, where the amplitude (y) is given by:y=sin[(mt+w)t]=sin(mt ² +wt)where

-   -   m is the linear rate of change in rotational speed,    -   w is the first order turning frequency, and    -   t is the time.

For each complete cycle of the sine waveform 412 in FIG. 4, or in otherwords wheremt ₁ ² +wt ₁=2π andmt ₂ ² —wt ₂=4π,

the amplitude is zero, which from above is expressed in an equation as:sin(mt ² +wt)=0.

Solving the two equations yields:w=2π(1−2t ₁ ² /t ₂ ²)/t ₁(1−t ₁ /t ₂), andm=(4π−wt ₂)/t ₂ ².

The two equations above are used to calculate frequency and linear rateof speed change, as given in block 508 of FIG. 5.

The rate of change m as initially calculated in this manner may not bethe same as that for the waveform in the adjacent block 1706 on theopposing side of the gap 1702, and therefore needs to be adjustedaccording to the method 218 a as depicted in FIG. 6. The process is toiteratively adjust the rate of change in rotational speed (m) until thefinal tachometer pulse in the gap 1702 matches the first tachometerpulse of the waveform in the adjacent block 1706, which is at t_(w).

Thus, the starting rotational speed rate of change m as initiallyestimated is either increased or decreased as necessary, depending on acomparison of the tachometer pulse widths t_(w) and t_(r), where t_(r)is the tach pulse width between the two tach pulses that are closest tothe gap 1702 in the reference block. The gap 1702 is then filled withtachometer pulses, where the final pulse within the gap 1702 Δt iscompared to t_(w). This process is repeated iteratively until theymatch.

Using the equation for a linear change in rotational speedy=sin{(mt+w)t}=sin(mt ² +wt).

The equation for the pulse times in the gap 1702 is given bysin(mt ² +wt)=0, whenmt ² +wt=2π_(I),

where (i) is the i^(th) tachometer pulse.

Solving this quadratic equation givest _(i) ={−w+√(w ²+8πm _(i))}/2m.

In regard to FIG. 6, method 218 a starts in block 604 by determining andsetting the variables that will be used in the method 218 a. Forexample, the time width of the gap 1702 t_(g) is determined, as well asthe time widths of the first tach pulse t_(w) in the adjacent block 1706and the last tach pulse t_(r) in the reference block 1704. A value isselected for an incremental amount Δm by which the speed m will beiteratively adjusted so as to make the tach pulses in the gap 1702 lineup with the tach pulses in the adjacent block 1706. A variable Δt isinitialized at the value of t_(r).

The time width between the two tach pulses that are closest to the gap1702 in the reference block 1704 and the two tach pulses that areclosest to the gap 1702 in the adjacent block 1706, t_(r) and t_(w)respectively, are compared to see which is greater, as given in block606. If t_(r) is less than t_(w), then the right-hand side of the method218 a as depicted in FIG. 6 is implemented in block 616. If t_(r) isgreater than t_(w), then the left-hand side of the method 218 a asdepicted in FIG. 6 is implemented in block 608.

In blocks 608 and 616, t_(w) is compared to Δt. In block 608, if Δt isnot greater than t_(w), then the method 218 a exits with the new speed mas previously determined. Similarly, in block 616, if Δt is greater thanor equal to t_(w), then the method 218 a exits with the new speed m aspreviously determined. However, if in either case the answer to the testof 608 or 616 is no, then the method falls respectively to block 612 or620, where variables are initialized for a loop where m is adjusteduntil the criteria as described herein have been met. In block 612, m istoo small, and so m is adjusted upwards by the value of Δm. In block620, m is too large, and so m is adjusted downwards by the value of Δm.

The time t is then tested against the gap time t_(g), as given in block614. If the test fails, then a new t is calculated as indicated in block622, and as described above. The new t is again tested against the gaptime t_(g), and when the new t eventually satisfies this condition inblock 614, the method 218 a returns blocks 608 and 616, where theiterations continue until the speed m is found which allows the tachsignals to align.

B. Linear Tach Pulses

Once the rate of change in rotational speed m has been adjusted, thenext step is to create the tachometer pulses in the gap 1702. This stepenables the gap 1702 to be filled with a representative wave 412.

Using the equation for a linear change in rotational speed shown above,the time for each tachometer pulse t_(i) is given bymt _(i) ² +wt _(i)=2π,

Solving this quadratic equation for t_(i) givest _(i) ={−w+√(w ²+8πm _(i))}/(2×m)

This is depicted as method 220 a in FIG. 7. The initial tach pulsecounter i is set to zero, as given in block 702. The time width of thegap 1702 t is derived, as given above and in block 704, and thetachometer pulse time t_(i) is calculated, also as given above and asindicated in block 706. The value of t_(i) is compared to t, as given inblock 708. If t_(i) is less than t, then there is room for another tachpulse, so the tach pulse number i is incremented as given in block 710,and the process returns to block 706 to calculate the next tach pulsetime t_(i). However, if t_(i) is not less than t, then all of the tachpulses have been created, and the method falls to block 712, and thetotal number of tach pulses created is the current value of i.

C. Linear Amplitude

Method 222 a depicted in FIG. 8 illustrates the process to calculate thelinear rate of change in waveform amplitude. The linear change inamplitude (y) is given byy=(1+at)×sin{(mt+w)t}

where a is the linear rate of change in waveform amplitude and t is thetime in seconds.

In FIG. 8, the RMS amplitude value (A_(r)) of the last cycle of thereference waveform 1704 is calculated, as given in block 802. The RMSvalue (A_(n)) of the first cycle of the adjacent waveform 1706 is alsocalculated, as given in block 804. The difference in the RMS amplitudevalues is converted to a peak value by multiplying by √2, as given inblock 806. The linear rate of change in waveform amplitude (a) iscalculated as given in block 808, usinga=(A _(n) −A _(r))*√2/T _(g)

where T_(g) is the time between the waveforms, or in other words, thegap 1702 time.

D. Linear Phase

The linear change in phase Δϕ is given byy=sin{(mt+w)t+Δϕt}

where Δϕ is the linear rate of change in waveform phase, and t is thetime in seconds. Method 224 a to accomplish this is depicted in FIG. 9.The change in phase ϕ_(r) of the last cycle of the reference waveform1704 is calculated using a Single Frequency Discrete Fourier Transform(SFDFT), as given in block 902. The waveform samples are taken betweenthe last two tachometer pulses in the reference waveform 1704, asexplained above.

The rate of change in phase is then estimated usingΔϕ_(r)=ϕ_(r) /t _(r)

as given in block 904, where t_(r) is the time between the twotachometer pulses. The change in phase ϕ_(w) of the first cycle of theadjacent waveform 1706 is calculated using a SFDFT, as given in block906. As before, the waveform samples are taken between the first twotachometer pulses in the adjacent waveform 1706. The rate of change inphase is then estimated as given in block 908, usingΔϕ_(w)=ϕ_(w) /t _(w)

where t_(w) is the time between the two tachometer pulses. The averagerate of change in phase for the waveform in the gap 1702 is thencalculated as given in block 910, usingΔϕ_(g)=(Δϕ_(r) +Δϕw)/2.E. Creation of a Linear Waveform in the Gap

The basic concept is to compress the last tachometer cycle of thereference waveform 1704 into the tachometer cycles in the gap 1702waveform, as given in methods 226 a and 226 c of FIGS. 10A and 10B,where t_(ref) represents the time interval in the reference waveform1704 and t_(gap) represents the time interval in the waveform gap 1702.

The waveform in the gap 1702 is then interpolated as described above inregard to FIG. 4 and method 226 a as given in FIG. 10A.

The amplitude of the wave is then adjusted for changes in amplitude andphase as given in flowcharts 222 a and 224 a. This process is repeateduntil the entire gap 1702 is filled.

This process is described in more detail with reference to methods 226 aand 226 b, as depicted in FIGS. 10A and 10B, respectively. First, theprocess variables are initialized, as started in block 1002, where theinitial tach pulse counter is set to zero, and then an incremental valueΔt is set as some small number, such as the sample interval divided by1,000, as given in block 1004. The frequency w_(r) and linear rate ofchange in frequency m for the reference waveform 1704 are initially usedin the calculations, as given in block 1006. Finally, t_(ref) andt_(gap) are initialized to values of zero, as given in block 1008.

The method then falls to a loop through method 226 c, as depicted inFIG. 10B, where the gap frequency is calculated as given in block 1022,the gap amplitude is calculated as given in block 1024, and thereference amplitude is calculated, as given in block 1026. If theamplitude of the reference block 1704 is less than the amplitude of thegap 1702 as determined in block 1028, then the reference time as used inthe calculations is incremented as given in block 1030, and thereference amplitude is recalculated as in block 1026. If the referenceamplitude is not less than the gap amplitude, then the referenceamplitude and time are found as given in block 1032, and y_(ref) isfound at t_(ref) through interpolation, as given in block 1034. The gap1702 amplitude is set to that value of y_(ref) as given in block 1036,and the gap 1702 amplitude is adjusted for growth as given in block1038. Finally, the gap 1702 amplitude is set to y_(ref) as given inblock 1040, and the method falls back to method 226 a as depicted inFIG. 10A.

After returning to the method 226 a, t_(gap) is incremented by thesample time, as given in block 1012, and it is determined whether thecycle of the currently reconstructed waveform is completed, as given inblock 1014. If the cycle of the current waveform is not completed, thenthe method returns to method 226 c. If the waveform cycle has beencompleted (one complete period), then the method falls to block 1016,where the value of i is incremented to the next tach pulse in the gap1702 waveform, for the possible creation of another waveform cycle byreturning to block 1008. However, if the last waveform was the finalwaveform to be placed into the gap 1702, as tested in block 1018, thenthe waveform reconstruction process is finished, as given in block 1020.

In some embodiments, the change in rotational speed across the gap 1702should not be more than about a factor of ten, as greater changes inrotational speed in some embodiments tends to reduce the number ofwaveform cycle samples to such an extent that detailed changes in thewaveform are lost.

In some embodiments, the change in rotational speed between the twowaveform cycles on either side of the gap 1702 is assumed to be linear.This is not normally an issue, as typically the change in rotationalspeed across the gap 1702 involves a large number of waveform cycles.

Numerical Solution for Exponential Section

A. Exponential Frequency and Speed

The process in this case is similar to the linear case, however, unlikethe linear case that had an analytical solution to find the frequencyand rate of change in rotational speed, the exponential change inrotational speed case requires a numerical iteration process to find thefrequency and change in rotational speed, as depicted in method 216 b ofFIG. 11.

As before, the last three tach pulses are taken from the reference block1704, as given in block 1104. However, in this case, the reference block1704 is, in some embodiments, the first waveform block following the gap1702, as it contains a waveform with a lower turning speed than theblock 1706 preceding the gap 1702, because the rotating machine isslowing down.

The tach pulse times are derived from the identified tach pulses, asgiven in block 1106, and an initial frequency is calculated, as given inblock 1108, based on a linear rate of change. However, the frequencydoes not change linearly in this section, as previously discussed, sothis calculated frequency is only a starting point for the followingcalculations. A small incremental value for adjusting this initialfrequency is selected, as given in block 1110, and the iterative processof bocks 1112 and 1114 are commenced.

The initial estimation of frequency uses the previously defined linearequationw=2π(1−2t ₁ ² /t ₂ ²)/t ₁(1−/t ₂)

With an exponential increase in frequency, the amplitude (y) is given byy=sin(wte ^(mt))sin(wte ^(mt))=0 whenwte ^(mt)=2π,4π, etc.

Taking the natural logarithm of each side of the equation givesln(wt ₁)+mt ₁=ln(2π) andln(wt ₂)+mt ₂=ln(4π)

Solving these two equations givesln(wt ₂)−t ₂ /t ₁ ln(wt ₁)=ln(4π)−t ₂ /t ₁ ln(2π), andm={ln(2π)−ln(wt ₁)}/t ₁

There is no analytical solution to the equation for w. Instead, w mustbe derived by a numerical iteration method of slowly increasing w fromits initial value as set using the linear equation above, until theequation is satisfied as depicted in block 1112 of flow chart 216 b ofFIG. 11. If it is not satisfied, then the calculated frequency isadjusted as given in block 1114, using the incremental amount set inblock 1110. Once w is derived, then the exponential change in rotationspeed m is calculated, as given in block 1116.

As before, the rate of change of m may not be the same as for theadjacent waveform 1706 across the gap 1702, and therefore may need to beadjusted as depicted in flowchart 218 b of FIG. 12. The process is toiteratively adjust the rate of change in rotational speed (m) until thefinal tachometer pulse width in the gap 1702 matches the first pulsewidth (t_(w)) of the adjacent waveform 1706. The starting rotationalspeed m is estimated as shown in FIG. 11, and is either increased ordecreased depending on the relative tachometer pulse width t_(w) andt_(r) as shown in FIG. 12. The gap 1702 is then filled with tachometerpulses, and the final pulse width in the gap 1702 Δt is compared tot_(w). This process is repeated iteratively until they match.

Using the equation for a linear change in rotational speedy=sin(wt ^(emt))

The equation for the pulse times in the gap 1702 is given bysin(wt ^(emt))=0 whenwt ^(emt)=2πi

where (i) is the i^(th) tachometer pulse.

This equation is solved iteratively by slowly increasing t untilwt ^(emt)>=2πi

In regard to FIG. 12, method 218 b starts in block 1204 by determiningand setting the variables that will be used in the method 218 b. Forexample, the time width t_(g) of the gap 1702 is determined, as well asthe time widths of the first tach pulse t_(w) in the adjacent block 1706and the last tach pulse t_(r) in the reference block 1704. A value isselected for an incremental amount Δm by which the speed m will beiteratively adjusted so as to make the tach pulses in the gap 1702 lineup with the tach pulses in the adjacent block 1706. A variable Δt isinitialized at the value of t_(r).

The time width between the two tach pulses that are closest to the gap1702 in the reference block 1704 and the two tach pulses that areclosest to the gap 1702 in the adjacent block 1706, t_(r) and t_(w)respectively, are compared to see which is greater, as given in block1206. If t_(r) is less than t_(w), then the right-hand side of themethod 218 b as depicted in FIG. 12 is implemented in block 1216. Ift_(r) is greater than t_(w), then the left-hand side of the method 218 bas depicted in FIG. 12 is implemented in block 1208.

In blocks 1208 and 1216, t_(w) is compared to Δt. In block 1208, if Δtis not greater than t_(w), then the method 218 b exits with the newspeed m as previously determined. Similarly, in block 1216, if Δt isgreater than or equal to t_(w), then the method 218 b exits with the newspeed m as previously determined. However, if in either case the answerto the test of 1208 or 1216 is no, then the method falls respectively toblock 1212 or 1220, where variables are initialized for a loop where mis adjusted until the criteria as described herein have been met. Inblock 1212, m is too small, and so m is adjusted upwards by the value ofΔm. In block 1220, m is too large, and so m is adjusted downwards by thevalue of Δm.

The time t is then tested against the gap time t_(g), as given in block1214. If the test fails, then t is tested in a different manner, asindicated in block 1222, and as described above. If that value of t doesnot meet the secondary criteria, which in some embodiments it would noton the first pass, then t is adjusted as given in block 1224, and thecriteria of block 1222 are again tested. When t eventually meets thecriteria of block 1222, then the values of the variables are set asgiven in block 1226, and the new t is again tested against the gap timet_(g). When the newt eventually satisfies the condition in block 1214,the method 218 b returns to the blocks 1208 and 1216, where theiterations continue until the speed m is found that allows the tachsignals to align.

B. Exponential Tach Pulses

Once the rate of change in rotational speed m has been adjusted, thenext step is to create the tachometer pulses in the waveform gap 1702.Using the equation for an exponential change in rotational speed, shownabove, the time for each tachometer pulse t_(i) is given bywt _(i) ^(emti)=2πi

There is no analytical solution to this equation for t_(i). In thiscase, t_(i) is derived by a numerical iteration method of slowlyincreasing t_(i) from a small initial value until the equation issatisfied as shown in flow chart 220 b of FIG. 13.

This is depicted as method 220 b in FIG. 13. The initial tach pulsecounter i is set to zero, as given in block 1302. The time t of thefirst tach pulse of the reference block 1704 is derived, as given inblock 1304 and above, and an incremental time Δt is set based on thesample interval, as given in block 1306. The time t is then incrementedby Δt until the condition specified in block 1308 is satisfied, and thenthe tachometer pulse time T_(i) is set to t, as indicated in block 1310.The value of T_(i) is compared to t, as given in block 1312. If T_(i) isless than t, then there is room for another tach pulse, so the tachpulse number i is incremented as given in block 1314, and the processreturns to block 1308 to calculate the next tach pulse time T_(i).However, if T_(i) is not less than t, then all of the tach pulses havebeen created, and the method falls to block 13162, and the total numberof tach pulses created is the current value of i.

C. Exponential Amplitude

In this case, the exponential increase in amplitude (y) is given byy=e ^(at)×sin(wte ^(mt))

where

a is the exponential rate of change in waveform amplitude, and

t is time in seconds

This is graphically illustrated in flowchart 222 b of FIG. 14. In block1402, the Root Mean Square (RMS) value (Ar) of the last cycle of thereference waveform 1704 is calculated. Then, the RMS value (An) of thefirst cycle of the adjacent waveform 1706 is calculated, as given inblock 1404. The RMS values are converted to peak values by multiplyingby √2, as given in block 1406. Finally, the linear rate of change in thewaveform amplitude (a) is computed as depicted in block 1408, by using:a=(ln(An*√2)−ln(Ar*√2)/T _(g)

where T_(g) is the time between the waveforms (i.e. the waveform gaptime).

D. Exponential Phase

The linear change in phase Δϕ is given by:y=sin(wte ^(mt) +e ^(Δϕt)),

where

Δϕ is the linear rate of change in waveform phase, and

t is the time in seconds.

With reference now to the method 224 b as depicted in FIG. 15, the firststep as given in block 1502 is to calculate the change in phase ϕ_(r) ofthe waveform cycle that is closest to the gap 1702 in the referenceblock 1704, using a Single Frequency Discrete Fourier Transform (SFDFT).These waveform samples are taken between the last two tachometer pulsesin the reference block waveform. The rate of change in phase is thenestimated as given in block 1504 by using:Δϕ_(r)=ϕ_(r) /t _(r)

where t_(r) is the time between the two tachometer pulses.

As given in block 1506, the next step is to calculate the change inphase ϕ_(w) of the first cycle of the adjacent waveform 1706 using aSFDFT. The waveform samples are taken between the two tachometer pulsesin the adjacent waveform 1706 that are closest to the gap 1702. The rateof change in phase is then estimated as given in block 1508 by using:Δϕ_(w)=ϕ_(w) /t _(w)

where t_(w) is the time between the two tachometer pulses.

For an exponential change in phase, the average rate of change in phasefor the waveform in the gap 1702 is then calculated as given in block1510, by using:Δϕ_(g)=(Δϕ_(r)+Δϕ_(w))/2

A small incremental phase adjustment δϕ is set, as given in block 1512.This is the amount by which the phase will be iteratively adjusted insubsequent steps, according to the numerical method. Then, as given inblock 1514, the phase Φ_(g) is calculated at t_(g) and t_(w), from whichΔΦ is calculated. ΔΦ is then tested against ϕ_(w), as given in block1516. If ΔΦ does not equal ϕ_(w), then Δϕ_(g) is adjusted by δϕ, asgiven in block 1518, and the iteration returns to block 1514. However,if ΔΦ does equal ϕ_(w), then that current value of Δϕ_(g) is taken asthe exponential phase change rate for the gap 1702, as given in block1520.

E. Creation of an Exponential Waveform in the Gap

The basic concept is to compress the nearest tachometer cycle of thereference waveform 1704 into the tachometer cycles in the gap 1702waveform, as given in methods 226 b and 226 d of FIGS. 16A and 16B,where t_(ref) represents the time interval in the reference waveform1704 and t_(gap) represents the time interval in the waveform gap 1702.

The waveform in the gap 1702 is then interpolated as described above inregard to FIG. 4 and method 226 b as given in FIG. 16A.

The amplitude of the wave is then adjusted for changes in amplitude andphase as given in flowcharts 222 b and 224 b. This process is repeateduntil the entire gap 1702 is filled.

This process is described in more detail with reference to methods 226 band 226 d, as depicted in FIGS. 16A and 16B, respectively. First, theprocess variables are initialized, as started in block 1602, where theinitial tach pulse counter is set to zero, and then an incremental valueΔt is set as some small number, such as the sample interval divided by1,000, as given in block 1604. The frequency w_(r) and linear rate ofchange in frequency m for the reference waveform 1704 are initially usedin the calculations, as given in block 1606. Finally, t_(ref) andt_(gap) are initialized to values of zero, as given in block 1608.

The method then falls to a loop through method 226 d, as depicted inFIG. 16B, where the gap frequency is calculated as given in block 1622,the gap amplitude is calculated as given in block 1624, and thereference amplitude is calculated, as given in block 1626. If theamplitude of the reference block 1704 is less than the amplitude of thegap 1702 as determined in block 1628, then the reference time as used inthe calculations is incremented as given in block 1630, and thereference amplitude is recalculated as in block 1626. If the referenceamplitude is not less than the gap amplitude, then the referenceamplitude and time are found as given in block 1632, and y_(ref) isfound at t_(ref) through interpolation, as given in block 1634. The gap1702 amplitude is set to that value of y_(ref) as given in block 1636,and the gap 1702 amplitude is adjusted for growth as given in block1638. Finally, the gap 1702 amplitude is set to y_(ref) as given inblock 1640, and the method falls back to method 226 b as depicted inFIG. 16A.

After returning to the method 226 b, t_(gap) is incremented by thesample time, as given in block 1612, and it is determined whether thecycle of the currently reconstructed waveform is completed, as given inblock 1614. If the cycle of the current waveform is not completed, thenthe method returns to method 226 d. If the waveform cycle has beencompleted (one complete period), then the method falls to block 1616,where the value of i is incremented to the next tach pulse in the gap1702 waveform, for the possible creation of another waveform cycle byreturning to block 1608. However, if the last waveform was the finalwaveform to be placed into the gap 1702, as tested in block 1618, thenthe waveform reconstruction process is finished, as given in block 1620.

In some embodiments, the change in rotational speed across the gap 1702should not be more than about a factor of ten, as greater changes inrotational speed in some embodiments tends to reduce the number ofwaveform cycle samples to such an extent that detailed changes in thewaveform are lost.

In some embodiments, the change in rotational speed between the twowaveform cycles on either side of the gap 1702 is assumed to beexponential. This is not normally an issue, as typically the change inrotational speed across the gap 1702 involves a large number of waveformcycles.

The foregoing description of embodiments for this invention has beenpresented for purposes of illustration and description. It is notintended to be exhaustive or to limit the invention to the precise formdisclosed. Obvious modifications or variations are possible in light ofthe above teachings. The embodiments are chosen and described in aneffort to provide illustrations of the principles of the invention andits practical application, and to thereby enable one of ordinary skillin the art to utilize the invention in various embodiments and withvarious modifications as are suited to the particular use contemplated.All such modifications and variations are within the scope of theinvention as determined by the appended claims when interpreted inaccordance with the breadth to which they are fairly, legally, andequitably entitled.

The invention claimed is:
 1. A method for filling a gap of missingvibration data in a set of vibration data, the method comprising thesteps of: sensing vibration data from a rotating machine with a sensor,creating the set of vibration data from the vibration data, reading theset of vibration data with a computer, selecting, with the computer, atleast one reference waveform on a first side of the gap and at least oneadjacent waveform on an opposing second side of the gap, determining,with the computer, whether the gap is in a section of the vibration datawhere a state of frequency of the vibration data is determined to be oneof changing linearly, changing exponentially, and steady state, when thegap is in a section of the vibration data where the state of thefrequency of the vibration data is changing linearly, applying, with thecomputer, an analytical method to at least one of the at least onereference waveform and the at least one adjacent waveform to approximatethe vibration data that is missing in the gap, when the gap is in asection of the vibration data where the state of the frequency of thevibration data is changing exponentially, applying, with the computer, anumerical method to at least one of the at least one reference waveformand the at least one adjacent waveform to approximate the vibration datathat is missing in the gap, when the gap is in a section of thevibration data where the state of the frequency of the vibration data issteady state, copying, with the computer, at least one of the at leastone reference waveform and the at least one adjacent waveform toapproximate the vibration data that is missing in the gap, andpresenting, with a display, the approximated vibration data with the gapfilled as a plot to a user.
 2. The method of claim 1, wherein when thegap is in a section of the vibration data where the state of thefrequency of the vibration data is changing, the at least one referencewaveform has a frequency that is slower than the at least one adjacentwaveform.
 3. The method of claim 1, wherein the set of vibration data issensed from a turbine.
 4. The method of claim 3, wherein the state ofthe frequency of the vibration data is changing linearly as a rotatingspeed of the turbine is increased to an operating speed.
 5. The methodof claim 3, wherein the state of the frequency of the vibration data ischanging exponentially as a rotating speed of the turbine is decreasedfrom an operating speed.
 6. The method of claim 3, wherein the state ofthe frequency of the vibration data is steady state as the turbine isrunning at an operating speed.
 7. The method of claim 1, wherein thestep of approximating the data using either the numerical method or theanalytical method further comprises: approximating a frequency rate ofchange of the missing vibration data in the gap, approximating tachsignal locations of the missing vibration data in the gap, approximatingan amplitude rate of change of the missing vibration data in the gap,approximating a phase rate of change of the missing vibration data inthe gap, filling the gap with ideal waveforms between the tach signallocations, and adjusting the ideal waveforms using the approximatedfrequency, amplitude, and phase rates of change.
 8. A method for fillinga gap of missing vibration data in a set of vibration data, the methodcomprising the steps of: sensing vibration data from a rotating machinewith a sensor, creating the set of vibration data from the vibrationdata, reading the set of vibration data with a computer, selecting, withthe computer, at least one reference waveform on a first side of the gapand at least one adjacent waveform on an opposing second side of thegap, determining, with the computer, whether the gap is in a section ofthe vibration data where a state of frequency of the vibration data isdetermined to be one of changing linearly, changing exponentially, andsteady state, when the gap is in a section of the vibration data wherethe state of the frequency of the vibration data is changing linearly,applying, with the computer, an analytical method to at least one of theat least one reference waveform and the at least one adjacent waveformto approximate the vibration data that is missing in the gap, when thegap is in a section of the vibration data where the state of thefrequency of the vibration data is changing exponentially, applying,with the computer, a numerical method to at least one of the at leastone reference waveform and the at least one adjacent waveform toapproximate the vibration data that is missing in the gap, when the gapis in a section of the vibration data where the state of the frequencyof the vibration data is steady state, copying, with the computer, atleast one of the at least one reference waveform and the at least oneadjacent waveform to approximate the vibration data that is missing inthe gap, wherein the step of approximating the data using either thenumerical method or the analytical method further comprises,approximating, with the computer, a frequency rate of change of themissing vibration data in the gap, approximating, with the computer,tach signal locations of the missing vibration data in the gap,approximating, with the computer, an amplitude rate of change of themissing vibration data in the gap, approximating, with the computer, aphase rate of change of the missing vibration data in the gap, andfilling, with the computer, the gap with ideal waveforms between thetach signal locations, and adjusting, with the computer, the idealwaveforms using the approximated frequency, amplitude, and phase ratesof change, and presenting, with a display, the approximated vibrationdata with the gap filled as a plot to a user.
 9. The method of claim 8,wherein when the gap is in a section of the vibration data where thestate of the frequency of the vibration data is changing, the at leastone reference waveform has a frequency that is slower than the at leastone adjacent waveform.
 10. The method of claim 8, wherein the set ofvibration data is sensed from a turbine.
 11. The method of claim 10,wherein the state of the frequency of the vibration data is changinglinearly as a rotating speed of the turbine is increased to an operatingspeed.
 12. The method of claim 10, wherein the state of the frequency ofthe vibration data is changing exponentially as a rotating speed of theturbine is decreased from an operating speed.
 13. The method of claim10, wherein the state of the frequency of the vibration data is steadystate as the turbine is running at an operating speed.